Contents

Idea

There are various different paradigms for the interpretation of predicate logic in type theory. In “logic-enriched type theory”, there is a separate class of “propositions” from the class of “types”. But we can also identify propositions with particular types. In the propositions as types-paradigm, every proposition is a type, and also every type is identified with a proposition (the proposition that it is an inhabited type).

Definition

In type theory

In homotopy type theory

For AA a type, the support of AA denoted supp(A)supp(A) or isInhab(A)isInhab(A) or τ−1A\tau_{-1} A or ‖A‖−1\| A \|_{-1} or ‖A‖\| A \| or, lastly, [A][A], is the higher inductive type defined by the two constructors

The recursion principle for supp(A)supp(A) says that if BB is a mere proposition and we have f:A→Bf: A \to B, then there is an induced g:supp(A)→Bg : supp(A) \to B such that g(isinhab(a))≡f(a)g(isinhab(a)) \equiv f(a) for all a:Aa:A. In other words, any mere proposition which follows from (the inhabitedness of) AA already follows from supp(A)supp(A). Thus, supp(A)supp(A), as a mere proposition, contains no more information than the inhabitedness of AA.